Rayleigh problem

In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of a plane from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations.

Flow description[1][2]

Consider an infinitely long plate which suddenly made to move with constant velocity in the direction, which is located at in an infinite domain of fluid, which is at rest initially everywhere. The incompressible Navier-Stokes equations reduce to

where is the kinematic viscosity. The initial and the no-slip condition on the wall are

the last condition is due to the fact that the motion at is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient.

Self-Similar solution[3]

The problem on the whole is similar to the one dimensional heat conduction problem. Hence a self-similar variable can be introduced

Substituting this the partial differential equation, reduces it to ordinary differential equation

with boundary conditions

The solution to the above problem can be written in terms of complementary error function

The force per unit area exerted on the plate by the fluid is

See also

References

  1. ↑ Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.
  2. ↑ Lagerstrom, Paco Axel. Laminar flow theory. Princeton University Press, 1996.
  3. ↑ Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.
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